Find the value of \(\left[\left(\frac{7}{6}\right)^{-9}\times \left(\frac{6}{7}\right)^{-5}\times \left(\frac{1}{8}\right)^{-4}\right]^{-5}\)

Given:

\(\left[\left(\dfrac{7}{6}\right)^{-9} \times \left(\dfrac{6}{7}\right)^{-5} \times \left(\dfrac{1}{8}\right)^{-4}\right]^{-5} \)

Formula used:

\(\left(\dfrac{a}{b}\right)^{-n} = \left(\dfrac{b}{a}\right)^{n} \)

Calculation:

\(\left[\left(\dfrac{7}{6}\right)^{-9} \times \left(\dfrac{6}{7}\right)^{-5} \times \left(\dfrac{1}{8}\right)^{-4}\right]^{-5} \)

⇒ \(\left[\left(\dfrac{6}{7}\right)^{9} \times \left(\dfrac{7}{6}\right)^{5} \times 8^{4}\right]^{-5} \)

⇒ \(\left[\left(\dfrac{6^{9}}{7^{9}}\right) \times \left(\dfrac{7^{5}}{6^{5}}\right) \times 8^{4}\right]^{-5} \)

⇒ \(\left[\dfrac{6^{9-5}}{7^{9-5}} \times 8^{4}\right]^{-5} \)

⇒ \(\left[\dfrac{6^{4}}{7^{4}} \times 8^{4}\right]^{-5} \)

⇒ \(\left[\dfrac{6^{4} \times 8^{4}}{7^{4}}\right]^{-5} \)

⇒ \(\left[\dfrac{(6 \times 8)^{4}}{7^{4}}\right]^{-5} \)

⇒ \(\left[\dfrac{48^{4}}{7^{4}}\right]^{-5} \)

⇒ \(\left(\dfrac{48}{7}\right)^{4 \times -5} \)

⇒ \(\left(\dfrac{48}{7}\right)^{-20} \)

⇒ \(\left(\dfrac{7}{48}\right)^{20} \)

∴ The correct answer is option (2).